Solve Using Elimination.${ \begin{align*} -8x + 3y &= 4 \ 4x + 4y &= 20 \end{align*} }$

Introduction

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are multiples of each other. In this article, we will use the elimination method to solve a system of two linear equations with two variables.

The Problem

We are given the following system of linear equations:

{ \begin{align*} -8x + 3y &= 4 \\ 4x + 4y &= 20 \end{align*} \}

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to make the coefficients of either $x$ or $y$ the same in both equations. We can do this by multiplying one or both of the equations by necessary multiples.

Let's multiply the first equation by 1 and the second equation by 2:

{ \begin{align*} -8x + 3y &= 4 \\ 8x + 8y &= 40 \end{align*} \}

Step 2: Add or Subtract the Equations

Now that we have the coefficients of $x$ in both equations, we can add or subtract the equations to eliminate $x$. Let's add the two equations:

{ \begin{align*} -8x + 3y + 8x + 8y &= 4 + 40 \\ 11y &= 44 \end{align*} \}

Step 3: Solve for the Variable

Now that we have eliminated $x$, we can solve for $y$ by dividing both sides of the equation by 11:

{ \begin{align*} y &= \frac{44}{11} \\ y &= 4 \end{align*} \}

Step 4: Substitute the Value of the Variable into One of the Original Equations

Now that we have found the value of $y$, we can substitute it into one of the original equations to find the value of $x$. Let's substitute $y = 4$ into the first equation:

{ \begin{align*} -8x + 3(4) &= 4 \\ -8x + 12 &= 4 \end{align*} \}

Step 5: Solve for the Variable

Now that we have substituted the value of $y$ into one of the original equations, we can solve for $x$ by subtracting 12 from both sides of the equation and then dividing both sides by -8:

{ \begin{align*} -8x &= 4 - 12 \\ -8x &= -8 \\ x &= \frac{-8}{-8} \\ x &= 1 \end{align*} \}

Conclusion

In this article, we used the elimination method to solve a system of two linear equations with two variables. We multiplied the equations by necessary multiples, added or subtracted the equations to eliminate one of the variables, solved for the variable, and then substituted the value of the variable into one of the original equations to find the value of the other variable. The final solution is $x = 1$ and $y = 4$.

Example Use Cases

The elimination method can be used to solve a wide range of systems of linear equations, including:

• Systems of two linear equations with two variables
• Systems of three linear equations with three variables
• Systems of linear equations with more than three variables

Advantages of the Elimination Method

The elimination method has several advantages, including:

• It is a simple and straightforward method to use
• It can be used to solve systems of linear equations with more than two variables
• It can be used to solve systems of linear equations with complex coefficients

Disadvantages of the Elimination Method

The elimination method also has several disadvantages, including:

• It can be time-consuming to use, especially for large systems of linear equations
• It can be difficult to determine which equations to add or subtract to eliminate one of the variables
• It can be difficult to solve for the variables if the coefficients of the variables are not multiples of each other.

Conclusion

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.

Q: When should I use the elimination method?

A: You should use the elimination method when the coefficients of the variables in the two equations are multiples of each other.

Q: How do I know which equations to add or subtract?

A: To determine which equations to add or subtract, you need to make the coefficients of either x or y the same in both equations. You can do this by multiplying one or both of the equations by necessary multiples.

Q: What if the coefficients of the variables are not multiples of each other?

A: If the coefficients of the variables are not multiples of each other, you may need to use a different method, such as the substitution method or the graphing method.

Q: Can I use the elimination method to solve systems of linear equations with more than two variables?

A: Yes, you can use the elimination method to solve systems of linear equations with more than two variables. However, it can be more complex and time-consuming to use.

Q: What are some common mistakes to avoid when using the elimination method?

A: Some common mistakes to avoid when using the elimination method include:

• Not making the coefficients of either x or y the same in both equations
• Not multiplying the equations by necessary multiples
• Not adding or subtracting the equations correctly
• Not solving for the variables correctly

Q: How do I know if I have solved the system of linear equations correctly?

A: To know if you have solved the system of linear equations correctly, you need to check your solution by substituting the values of the variables back into the original equations.

Q: What are some real-world applications of the elimination method?

A: The elimination method has many real-world applications, including:

• Solving systems of linear equations in physics and engineering
• Solving systems of linear equations in economics and finance
• Solving systems of linear equations in computer science and data analysis

Q: Can I use the elimination method to solve systems of linear equations with complex coefficients?

A: Yes, you can use the elimination method to solve systems of linear equations with complex coefficients. However, it can be more complex and time-consuming to use.

Q: What are some tips for using the elimination method effectively?

A: Some tips for using the elimination method effectively include:

• Make sure to make the coefficients of either x or y the same in both equations
• Multiply the equations by necessary multiples
• Add or subtract the equations correctly
• Solve for the variables correctly
• Check your solution by substituting the values of the variables back into the original equations

Conclusion

In conclusion, the elimination method is a powerful technique for solving systems of linear equations. It is a simple and straightforward method to use, and it can be used to solve systems of linear equations with more than two variables. However, it can be time-consuming to use, and it can be difficult to determine which equations to add or subtract to eliminate one of the variables. By following the tips and avoiding the common mistakes, you can use the elimination method effectively to solve systems of linear equations.